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Tamarin: Principles of Genetics, Seventh Edition
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IV. Quantitative and Evolutionary Genetics
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19. Population Genetics: The Hardy Weinberg Equilibrium and Mating Systems
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The McGraw Hill Companies, 2001
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Table 19.5 Genotypic Proportions in a Population with Inbreeding
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Genotype AA Aa aa Total (p2 Due to Random Mating (1 F) p 2 (1 2pq (1 q (1 2pq (1
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Due to Inbreeding (F) pF qF
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Observed Proportions p2 2pq(1 q
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F) F) F) q 2 ) (1 F) F) (p
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cousins are the offspring of siblings, they share a set of common grandparents. Thus, individual I can be autozygous for alleles from either ancestor A or B, her greatgrandparents.The path diagram shows the only routes by which autozygosity can occur. The inbreeding coef cient of the offspring of rst cousins can be calculated as follows.The path diagram of gure 19.3 is shown again in gure 19.4, with lowercase letters designating gametes. Two paths of autozygosity appear in this diagram, one path for each grandparent as a common ancestor: A to D and E, then to G and H, and nally to I; or B to D and E, then to G and H, and nally to I. In the path with A as the common ancestor, A contributes a gamete to D and a gamete to E. The probability is one-half that D and E each carry a copy of the same allele. That is, there are four possible allelic combinations for the two gametes, a1 and a2: A-A; A-a; a-A; and a-a. Of these combinations, the rst and last (A-A and a-a) give a copy of the same allele to the two offspring, D and E, and can thus contribute to autozygosity. The probability that gametes a1 and d carry copies of the same allele is onehalf, and the probability that d and g carry copies of the same allele is also one-half. Similarly, on the other side of the pedigree, the probability is one-half that a2 and e carry copies of the same allele and one-half that e and h carry copies of the same allele. Thus, the overall probability that the alleles that g and h carry are identical by descent (autozygous) is (1/2)5. In general, it would be (1/2)n for each path, where n is the number of ancestors in the path. You may have spotted an additional factor here. Of the possible combinations of allelic copies passed on to D and E, one-half (A-A and a-a) are autozygous combinations. However, the other half of the combinations, A-a and a-A, can lead to autozygosity if A is itself inbred. If we let FA be the inbreeding coef cient of A (the probability that any two alleles at a locus in A are identical by descent), then FA is the probability that the A-a and a-A combinations are also autozygous. Thus, the probability that a common ancestor, A, passes on copies of an
A A B
D C D E F G G H
I Pedigree
I Path diagram
Figure 19.3 Conversion of a pedigree to a path diagram. This pedigree depicts the mating of rst cousins. In the path diagram, all extraneous individuals are removed, leaving only those who could contribute to the inbreeding of individual I. Individuals in the line of descent are connected directly with straight lines, indicating the paths along which gametes are passed.
a2 b1 a1
I Figure 19.4
The path diagram of the mating of rst cousins with gametes labeled in lowercase letters.
Tamarin: Principles of Genetics, Seventh Edition
IV. Quantitative and Evolutionary Genetics
19. Population Genetics: The Hardy Weinberg Equilibrium and Mating Systems
The McGraw Hill Companies, 2001
Nineteen
Population Genetics: The Hardy-Weinberg Equilibrium and Mating Systems
identical ancestral allele is 1/2 (1/2)FA, or (1/2) (1 FA ). In other words, there is a one-half probability that the alleles transmitted from A to D and E are copies of the same allele. In the other half of the cases, these alleles can be identical if A is inbred. The probability of identity of A s two alleles is FA. The expression for the inbreeding coef cient of I, FI, can now be changed from (1/2)n by substituting (1/2)(1 FA ) for one of the (1/2)s to FI (1 2)n(1 FA )
Thus, about 25% of the loci in an offspring of siblings are autozygous.
Path Diagram Rules
The following points should be kept in mind when calculating an inbreeding coef cient: 1. All possible paths must be counted. A path is possible if gametes can actually pass in that direction. Paths that violate the rules of inheritance cannot be used. For example, in gure 19.4, the following path is unacceptable: I G E A D H I. 2. In any path, an individual can be counted only once. 3. Every path must have one and only one common ancestor. The inbreeding coef cient of any other individual in the path is immaterial. In gure 19.6, we present a complex pedigree produced from repeated sib mating, a pattern found in livestock and laboratory animals. This pedigree has two interesting points. First, common ancestors occur in several different generations. Second, some of the paths are complex. Thus, we must be sure to count all paths (paths 5 and 6 might not be immediately obvious). Although not shown in gure 19.6, one of the common ancestors, A, is also inbred (FA 0.05) a fact that we must take into consideration in paths 3 and 5. Thus, FI is as follows: From path 1: (1/2)3 From path 2: (1/2)
3 5 5 5 5
This equation accounts only for the inbreeding of I by the path involving the common ancestor, A, and does not account for the symmetrical path with B as the common ancestor. To obtain the total probability of inbreeding, the values from each path must be added (because these are mutually exclusive events; see chapter 4). Thus the complete formula for the inbreeding coef cient of the offspring of rst cousins is FI [(1 2) (1
FJ)]
(19.1)
in which FI is the probability that the two alleles in I are identical by descent, n is the number of ancestors in a given path, FJ is the inbreeding coef cient of the common ancestor of that path, and all paths are summed. In the example of the mating of rst cousins ( g. 19.4) FI (1 2)5(1 FA ) (1 2)5(1 FB )
If we assume that FA and FB are zero (which we must assume when the pedigrees of A and B are unknown), then FI 2(1 2)5 (1 2)4 0.0625
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